Kun Yang National High Magnetic Field Lab

and Florida State University

In Collaboration with Yafis Barlas (NHMFLàUC Riverside)

Hui Zhai (Tsinghua, Beijing)

Quantum Hall Effect: a Paradigm of Topological Order II

Topological Phase Transitions in Quantum Hall Systems

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B (T)

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1,2,3,n nhcn eBφ

ν = = = L

Integer Quantum Hall Effect

Question: How do transitions between plateaus occur?

Partial Answer: Quantum Percolation

Interplay of disorder and interaction not yet fully understood!

Outline •  Brief introduction of quantum control in cold atom

systems and some advertisements. •  A quantum Hall transition near an s-wave Feshbach

resonance in a rotating fermion gas, and emergent quantum particles obeying semionic statistics at the quantum phase transition. (KY and H. Zhai, PRL 08).

•  A quantum Hall transition near a p-wave Feshbach resonance in a rotating fermion gas, and fractionalization via a Z2 gauge field or visons. (Y. Barlas and KY PRL 11).

•  Summary.

Quantum Control in Cold Atom Systems •  Atoms/molecules with both Fermi and Bose statistics,

and in particular their mixtures are available (can be used to realize supersymmetry! Y. Yu and KY, PRLs 08 and 10).

•  Optical lattice potential can be controlled through frequency and intensity of light; band structure/effective mass controllable (achieve boson superfluid-insulator transition; possible new universality class in the presence of fermions. KY PRB 08).

•  “Magnetic field” controlled through rotation of optical lattice/trap (Coriolis’ force “ ≈ ” Lorentz force), or synthetic gauge field (Spielmann).

•  Interaction strength (and sign!) controlled via (real) magnetic field tuned Feshbach resonances.

Preliminary Evidence of FQH-like Correlation Reported Recently in Rotating Optical Lattices!

Nathan Gemelke, Edina Sarajlic, and Steven Chu, arxiv1007

Vortices observed in a synthetic gauge field (Lin et al., Nature 10)

Quantum Hall States in a Rotating Trap •  Integer quantum Hall (IQH) states: fully-filled

Landau levels, filling factor ν = m; fermions only. •  Fractional quantum Hall (FQH) states: Laughlin

type of states with ν = 1/m; even m for boson and odd m for fermion.

•  Starting with spin-1/2 fermionic atom IQH state at ν = 2; turning on strong s-wave attractionè bosonic molecule FQH state at ν = ½! (Haldane+Rezayi 04) Same quantized Hall conductance (e*: atom “charge”):

Same quasiparticle charge: e*=2e*/2

•  Spin QH state. •  Quasiparticles are

spin-1/2 fermions. •  Two edge modes.

•  Spin insulator. •  Quasiparticles are

spinless semions. •  One edge mode.

There must be a quantum phase transition between boson FQH and fermion IQH states!

KY and H. Zhai, PRL 08

Chern-Simons-Ginzburg-Landau (CSGL) Theory

Conserved charges:

Constraints from CS term:

Mean-field approximation:

Mean-field Description of Phases Within mean-field treatment of statistical flux attached to particles, Quantum Hall Effect = Bose condensation of Chern-Simons bosons! (Zhang, Hansson and Kivelson 89; Read 89)

Atomic IQH phase: two condensates and both U(1) symmetries broken:

Molecular FQH phase: one condensate and only one U(1) symmetry broken:

Mean-field Description of Phase Transition

Bogoliubov transformation:

Integrate out massive field :

Effective theory is that of 2+1D XY transition with Lorentz invariance! But coupling to CS field must be included.

Full Theory of QH Phase Transition Perform a similar transformation on the CS gauge field:

Integrate out massive field :

Theory describes condensation of emergent particles with zero charge, spin-1/2, and semion statistics!

Comments on the Effective Field Theory

•  Properties (or quantum numbers) of emergent particle expected from difference between two phases.

•  Same types of theory studied earlier in context of QH-Insulator transition on a lattice; nature of transition controversial: Large-N limit suggests 2nd order transition with θ’-dependent critical behavior (Wen and Wu 92, Chen, Fisher and Wu 93); while Pryadko and Zhang (94) found fluctuation-driven 1st order transition in certain parameter range.

•  Order of transition resolvable in cold atom system by measuring spin-gap using RF absorption, or numerical simulation (Haldane and Rezayi). Not (yet) realizable in electronic systems.

A Simpler (but More Interesting) Case

Starting with spinless fermionic atom IQH state at ν = 1; turning on strong p-wave pairing interaction è bosonic molecule FQH state at ν = ¼ !

Same quantized Hall conductance (e*: atom “charge”):

Different quasiparticle charge: e* ≠ 2e*/4=e*/2; charge fractionalization! Different statistcs as well. Same number of edge modes; similar edge physics!

CSGL Theory of Phases and Phase Transition

Differences from previous case: • Only one Chern-Simons Gauge Field (coupled to charge). • Only one condensate in both phases! • Gauge field has an Anderson-Higgs mass (overwhelming Chern-Simons term), and plays no-role at the transition! • Reduces to an Ising model:

• Stable 2nd order transition; unlike transition between atomic and molecular BECs unstable due to collapse of system near Feshbach resonance. • Critical properties known accurately: z=1, υ≈0.63; neutral gap vanishing with exponent υz near critical point.

• However: Looks like a conventional (Landau-type) transition; topological nature of phases involved and transition not explicit. • Change of topological properties not manifested, like change of torus degeneracy, charge fractionalization etc.

Theory identical to that of transition between atomic and molecular BECs without rotation (Radzihovsky+Park +Weichman; Romans+Duine+Sachdev+Stoof, PRLs 04); Ising transition!

(Dual) Chern-Simons/Z2 Gauge Theory

Conserved atom 3-current.

σ =± 1: Z2 gauge field. φ: phase of molecular (half) vortex field.

Phases and Phase Transition: • Small β: Z2 vortices (or visons) condensed; molecular vortices confined; only atomic vortices present, which represent original charge-1, fermionic quasiparticles. IQH phase. • Large β: Z2 vortices (or visons) gapped; molecular vortices de-confined = Laughlin quasiparticle; fractionalization! FQH phase. • At transition: vison gap closes; Ising criticality.

Non-trivial checks: • Torus degeneracy changes from 1 to 4 due to the phase transition in Z2 sector. Topological nature of phases and phase transition explicitly manifested. • No additional edge modes!

Story similar to, but simpler than Senthil-Fisher theory for spin-charge separation in cuprates.

(See Y. Barlas and KY, PRL 11 for more details)

Summary/Conclusions • Trapped cold atom systems offer opportunities to study strongly correlated systems different from those encountered in electronic systems. • Discussed examples of quantum Hall phase transitions driven by attractive interactions, whose critical theories are relativistic massless semions, and massless visons in Z2

gauge theories respectively. These transitions are very similar to quantum number fractionalization transitions discussed in other contexts, but have not yet found concrete experimental realizations there. • Represent two large classes of Abelian QH transitions: with or without additional condensates/edge modes. • Outlook: Possible transitions between Abelian and non-Abelian QH states, like υ=4 fermions to υ=1 bosons (Moore-Read).